Graphs of Large. Connectivity and. Chromatic Number. N. Alon DEPARTMENT OF MATHEMATICS TEL AVIV UNIVERSITY RAMAT AVIV. TEL AVIV ISRAEL

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1 Subgraphs of Large Connectivity and Chromatic Number in Graphs of Large Chromatic Number N. Alon DEARTMENT OF MATHEMATICS TEL AVIV UNIVERSITY RAMAT AVIV. TEL AVIV ISRAEL D. Kleitman DEARTMENT OF MATHEMATICS MASSACHUSETTS INS T/ TU TE OF TECHNOLOGY, CAMBRIDGE, MASSACHUSETTS ABSTRACT C. Thomassen MATHEMATICAL INSTITUTE, BLD. 303 TECHNlCAL UNlVERSITY OF DENMARK DK-2800 LYNGBY DENMARK M. Saks and. Seymour BELL COMMUNICATIONS RESEARCH 435 SOUTH STREET MORRISTOWN, NEW JERSEY For each pair k, rn of natural numbers there exists a natural number f(k, rn) such that every f(k, m)-chromatic graph contains a k-connected subgraph of chromatic number at least rn. Journal of Graph Theory, Vol. 11, No. 3, (1987) by John Wiley & Sons, Inc. CCC /87/ $04.00

2 368 JOURNAL OF GRAH THEORY INTRODUCTION Mader [I] proved that every graph of minimum degree at least 4k contains a k-connected subgraph. Thus every (4k + I)-chromatic graph contains a k-connected subgraph. In this note we show that a graph of sufficiently large chromatic number contains a subgraph that has both large connectivity and large chromatic number. This result, which is useful for finding general configurations in graphs of large chromatic number (see 131). was first stated in 121 but the proof given there is in error. We prove the following: Theorem. Every graph G of chromatic number greater than p = max (look3, 10k + m) contains a (k + 1)-connected subgraph of chromatic number at least m. NOTATION AND ROOF OF THE THEOREM For any vertex set A of G we denote by G(A) the subgraph of G induced by A and G - A denotes G[V(G)\A]. As usual x(g) denotes the chromatic number of G. The number of neighbors in A of a vertex u is denoted d,(u). The A-weight of u is defined as 2k w,(u) = 2k min[d,(u),p] and, for each vertex set S of G, we put where the summation is taken over all u in S. Note that w,,(s) 2 1 always. Finally, we put W = IOk. We now prove the theorem. Without loss of generality we can assume that G is ( p + 1)-color-critical and hence all vertices of G have degree at least p. If G is (k + 1)-connected, there is nothing to prove, so G has a separating vertex set S with at most k vertices. If A is the union of the (vertex sets of) some connected components of G - S then clearly Among all pairs S,A where S is a separating vertex set and A the union of some (but not all) vertex sets of connected components of G - S satisfying (l), we choose one such that IAI is minimal. We shall prove that G(A U S) has the desired properties. x[g(a US)] 2 m. (2)

3 SUBGRAHS OF LARGE CONNECTIVITY AND CHROMATIC NUMBER 369 roof of (2). Since G is (p + 1)-color-critical, x(g - A) 5 p. If x[g(a)] 5 p - \St. then any p-coloring of G - A can be extended to a p-coloring of G, which is impossible. So x[g(a)] > p - IS1 and, by (1) x[g(a U S)] 2 x[g(a)] 2 p - IS1 + 1 > 10k + in -!A = m. which proves (2). It remains to be shown that G(A U S ) is (k + I)-connected. We first prove an auxiliary result: For each u in S,d,,(u) 2 k + 1. (3) roof of (3) (by contradiction). Suppose that (I,(U) 5 k for some u in S. Let N be the set of neighbors of u in A. Since A is nonempty and G has minimum degree at least p, it follows that We put S = (S\{u}) U N and A = AW. Then 0 < IA I < IA I and, for every vertex u in N, Hence da.(u) 1) p - W - k + 1. Also 2k MJ,,.(S) - W,(S) 5 W-k Combining the last two inequalities we get 2k rw +W- -2k- 1

4 370 JOURNAL OF GRAH THEORY Hence the pair S,A satisfies (l), contradicting the minimality of \A\. This proves (3). G(A U S) is (k + 1)-connected. (4) roof of (4) (by contradiction). Suppose S is a separating vertex set of G(A U S) such that IS / 5 k. Then the vertex set of G(A U S ) - S can be partitioned into two nonempty sets A, U S, and A? U S2 such that there is no edge from A, U S, to A2 U S, and A, U A, C_ A, S, U S, C S. By (3), each of A,, A2 is nonenipty. Then each of S U S, and S U Sz is a separating vertex set of G and without loss of generality we can assume that In particular, w,(s,) 5 (W/2). Now W 2k W k(2k + 1) k + -- Hence the pair S U S,, A, satisfies (I), contradicting the minimality of \A\. This proves (4) and the theorem. The Theorem shows that f(k,m) 5 look3 + m, where f (k, m) is the (smallest) number satisfying the statement of thc abstract. We obtain the lower bound f(k,m) 2 k + m - 2 as follows: Take k - 1 disjoint copies of the complete graph Kk-l. For each vertex set S containing precisely one vertex of each Kk-] we add a K,,-: and join it completely to S. Then the resulting graph G.,,, has chromatic number k + m - 3 and no k-connected subgraph of GL,,,, contains vertices of two distinct K,,.2 - s. Hence every k-connected subgraph of GL,,n is (m- I)- colorable. References [I J W. Mader, Existenz n-fach zusammcnhiingender Teilgraphen in Graphen geniigend grossen Kantendichte, Abh. Math. Sem. Hamburg Univ. 37 (1972)

5 SUBGRAHS OF LARGE CONNECTIVITY AND CHROMATIC NUMBER 371 [2] C. Thomassen, Graph decomposition with applications to subdivisions and path systems modulo k. J. Graph Theory 7 (1983) [3] C. Thomassen, Configurations in graphs of large minimum degree, connectivity or chrcmatic number. To appear.